Finding the limit of diverging components in three-way Candecomp/Parafac—A demonstration of its practical merits

Abstract

Three-way Candecomp/Parafac (CP) is a three-way generalization of principal component analysis (PCA) for matrices. Contrary to PCA, a CP decomposition is rotationally unique under mild conditions. However, a CP analysis may be hampered by the non-existence of a best-fitting CP decomposition with R≥2 components. In this case, fitting CP to a three-way data array results in diverging CP components. Recently, it has been shown that this can be solved by fitting a decomposition with several interaction terms, using initial values obtained from the diverging CP decomposition. The new decomposition is called CPlimit, since it is the limit of the diverging CP decomposition. The practical merits of this procedure are demonstrated for a well-known three-way dataset of TV-ratings. CPlimit finds main components with the same interpretation as Tucker models or when imposing orthogonality in CP. However, CPlimit has higher joint fit of the main components than Tucker models, contains only one small interaction term, and does not impose the unnatural constraint of orthogonality. The uniqueness properties of the CPlimit decomposition are discussed in detail.